Equivariant K-theory classes of matrix orbit closures

Abstract

The group $G=G{L}_r(k)\times (k^{\times }{)}^n$ acts on ${\mathbf{A}}^{r\times n}$, the space of $r$-by-$n$ matrices: $G{L}_r(k)$ acts by row operations and $(k^{\times }{)}^n$ scales columns. A matrix orbit closure is the Zariski closure of a point orbit for this action. We prove that the class of such an orbit closure in $G$ equivariant $K$-theory of ${\mathbf{A}}^{r\times n}$ is determined by the matroid of a generic point. We present two formulas for this class. The key to the proof is to show that matrix orbit closures have rational singularities.

27pp. Expanded introduction. New section with positivity conjectures for all matroids. Comments welcome!

References in corpus (2)

• Equivariant K-theory of Grassmannians
• Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory

Cited by in corpus (1)

• Orbits in $({\mathbb{P}}^r{)}^n$ and equivariant quantum cohomology